# Ex 7.5, 15

Last updated at Dec. 8, 2016 by Teachoo

Last updated at Dec. 8, 2016 by Teachoo

Transcript

Ex 7.5, 15 1 𝑥4 − 1 Now, 1 𝑥4 − 1= 1 𝑥2 + 1 𝑥2 − 1 = 1 𝑥2 + 1 𝑥 − 1 𝑥 + 1 Hence we can write this as 1 𝑥2 + 1 𝑥 − 1 𝑥 + 1= 𝐴𝑥 + 𝐵 𝑥2 + 1 + 𝐶𝑥 − 1 + 𝐷𝑥 + 1 1 𝑥2 + 1 𝑥 − 1 𝑥 + 1 = 𝐴𝑥 + 𝐵 𝑥 − 1 𝑥 + 1 + 𝐶 𝑥2 + 1 𝑥 + 1 + 𝐷 𝑥2 + 1 𝑥 − 1 𝑥2 + 1 𝑥 − 1 𝑥 + 1 By cancelling denominator 1= 𝐴𝑥+𝐵 𝑥−1 𝑥+1+𝐶 𝑥2+1 𝑥+1+𝐷 𝑥2+1 𝑥−1 Putting x = 1 1= 𝐴𝑥+𝐵 𝑥−1 𝑥+1+𝐶 𝑥2+1 𝑥+1+𝐷 𝑥2+1 𝑥−1 1= 𝐴 1+𝐵 1−1 1+1+𝐶 12+1 1+1+𝐷 12+1 1−1 1= 𝐴+𝐵×0×2+𝐶 2× 2+𝐷 2×0 1=0+𝐶×4+0 1=4𝐶 𝐶= 14 Similarly putting x = −1, in (1) 1= 𝐴 −1+𝐵 −1−1 −1+1+𝐶 −12+1 −1+1+𝐷 −12+1 −1−1 1= −𝐴+𝐵 −2 0+𝐶 1+1 0+𝐷 1+1 −2 1=0+0+𝐷×2× −2 1=−4𝐷 𝐷= −14 Putting x = 0 , in (1) 1= 𝐴𝑥+𝐵 𝑥−1 𝑥+1+𝐶 𝑥2+1 𝑥+1+𝐷 𝑥2+1 𝑥−1 1= 𝐴×0+𝐵 0−1 0+1+𝐶 0+1 0+1+𝐷 0+1 0−1 1=𝐵 −1 1+𝐶 1 1+𝐷 1 −1 1=𝐵+𝐶−𝐷 1=−𝐵+ 14 − −14 1=−𝐵+ 12 𝐵= 12−1 𝐵= −12 Putting 𝑥= 2 in (1). 1= (A𝑥+B) (𝑥−1) + C ( 𝑥2+1) (𝑥+1) + D ( 𝑥2+1) (𝑥−1) 1 = (2A + B) (2 − 1) (2 + 1) + C ( 22+1) (2 +1) D ( 22+1) (2− 1) 1 = (2A + B) (1) (3) + C (5) (3) + D (5) (1) 1 = 6A + 3B + 15C +5D 1 = 6A − 32 + 154 − 54 1 = 6A − 32 + 104 1 = 6A + 1 6A = 1 − 1 6A = 0 A = 0 Therefore we can write 1 𝑥2 + 1 𝑥 − 1 𝑥 + 1= 𝐴𝑥 + 𝐵 𝑥2 + 1 + 𝐶𝑥 − 1 + 𝐷𝑥 + 1 1 𝑥4 − 1= 0 × 𝑥 + − 12 𝑥2 + 1 + 14𝑥 − 1 + −14𝑥 + 1 = −12 𝑥2 + 1 + 14 𝑥 − 1 − 14 𝑥 + 1 Integrating 𝑤.𝑟.𝑡.𝑥. 1 𝑥4 − 1 𝑑𝑥= −12 𝑥2 + 1 + 14 𝑥 − 1 − 14 𝑥 + 1 𝑑𝑥 = − 12 1 𝑥2 + 1 𝑑𝑥+ 14 1𝑥 − 1 𝑑𝑥− 14 1𝑥 + 1 𝑑𝑥 = −12 tan−1𝑥+ 14 log 𝑥−1− 14log 𝑥+1+ C = −12 tan−1𝑥+ 14 log 𝑥−1−log 𝑥+1]+ C = −𝟏𝟐 𝒕𝒂𝒏−𝟏𝒙+ 𝟏𝟒 𝐥𝐨𝐠 𝒙 − 𝟏𝒙 + 𝟏+ C

Ex 7.5, 1

Ex 7.5, 2

Ex 7.5, 3

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Ex 7.5, 5

Ex 7.5, 6

Ex 7.5, 7

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Ex 7.5, 9 Important

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Ex 7.5, 11 Important

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Ex 7.5, 15 You are here

Ex 7.5, 16

Ex 7.5, 17 Important

Ex 7.5, 18 Important

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Ex 7.5, 21 Important

Ex 7.5, 22

Ex 7.5, 23

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.